## Fréchet Distance in Subquadratic time

- Haoqiang Huang, HKUST
- Time: 2024-09-19 14:00
- Host: Turing Class Research Committee
- Venue: Room 204, Courtyard No.5, Jingyuan

### Abstract

Let m and n be the numbers of vertices of two polygonal curves in d-dimensional space for any fixed d such that m\le n. Since it was known in 1995 how to compute the Fréchet distance of these two curves in O(mnlog(mn)) time, it has been an open problem whether the running time can be reduced to o(n2) when m = \Omega(n). In the meantime, several well-known quadratic time barriers in computational geometry have been overcome: 3SUM, some 3SUM-hard problems, and the computation of some distances between two polygonal curves, including the discrete Fréchet distance, the dynamic time warping distance, and the geometric edit distance. It is curious that the quadratic time barrier for Fréchet distance still stands. We present an algorithm to compute the Fréchet distance in O(mn(log log n)2+_ log n= log1+_m) expected time for some constant _ 2 (0; 1). It is the first algorithm that returns the Fréchet distance in o(mn) time when m = \Omega(n^\epsilon) for any \epsilon in (0, 1].

### Biography

Haoqiang is a fifth-year PhD student at HKUST. He is broadly interested in theoretical computer science, including computational geometry, mechanism design, and online algorithms. His recent research focuses on problems involving trajectory/time series data analytic tasks including compression, clustering, and nearest neighbor retrieval.